What is the constant term in the expansion of \(\left(\sqrt{x}+\dfrac5x\right)^{9}\)?
+9519 Considering the general term of the expansion:
\(\left(\sqrt x + \dfrac5x\right)^9 = \displaystyle\sum_{k = 0}^9 \binom{9}k\cdot \left(\sqrt x\right)^k \left(\dfrac5x\right)^{9-k}\)
Simplifying,
\(\left(\sqrt x + \dfrac5x\right)^9 = \displaystyle\sum_{k = 0}^9 \binom{9}k \cdot 5^{9-k}\cdot x^{3k/2-9}\)
When the term is constant, power of x is 0.
\(\dfrac{3k}2 - 9 = 0\\ k = 6\)
Now, substitute k = 6 into the general term, you get the constant term.
(Note that if k was not an integer, the constant term would be 0.)
